Sum of the reciprocals of the squares of the prime numbers

krassi_holmz · 2006-02-18 22:44:56

Very interesting prblem:

krassi_holmz · 2006-02-18 22:46:53

We know that:

So
0<Sum<Pi^2/6

krassi_holmz · 2006-02-18 22:48:35

I'll try to make a program for the sum.

krassi_holmz · 2006-02-18 22:51:33

Here's my program:

k = 100000;
ss = 1000;
Print["Initialization Done"];
Do[prt[i] = Prime[i], {i, 1, k}];
Print["Prime Geneneration Done"];
Do[prtsq[i] = prt[i]*prt[i], {i, 1, k}];
Print["Squaring Done"];
Do[thearr[i] = 1/prtsq[i], {i, 1, 100000}];
Print["Factorising Done"];
sum = 0;
Do[sum += N[thearr[i], ss], {i, 1, 100000}];
Print["Sum = ", sum];

And here's what I get for the sum:
0.4522473688277700954422164277522326096645066956889359406373748543156987039966
525643794108225095161389758244363445339660776113109242885808492718205416638194
526958800379786519866057949036856229177825278987902389864970540352844167196863
834688645075159015805935496856212223538491039364902103635445065361656745970932
905021125274435235568533828552570131338540212094414196757580439057169807537575
899440924188480149221835203239871132457344235369007243117411236106046262308444
576868350724155932530725321698618605264299442562165096782086212268498382688102

krassi_holmz · 2006-02-18 22:52:49

Is this sum a irrational number?
Is It transcedent?
Can we compute Prime[n] from it?

John E. Franklin · 2006-02-20 09:09:29

I get everything except the N and the ss usage.

krassi_holmz · 2006-02-22 02:54:48

In Mathematica the "N(number,precise)" function gives the number with precisedigits after the demical point.
Here's an example:
N[Pi,2]=3.14
N[Pi,4]=3.1415
ss is the precise. Because ss=1000 we can be sure that more than 100 digits or this number are correct.

John E. Franklin · 2006-02-22 05:55:10

And if Prime[] returns 0 if not prime, then why don't you get divide by zero error?
Tell me about Prime[]
Does prime return infinitywhen not prime?

MathsIsFun · 2006-02-22 09:15:05

Maybe we could it "Krassi's Number"

krassi_holmz · 2006-02-28 06:15:37

For John, not, PrimeQ[] is for testing whatever a number is prime.
Prime[n] gives the n-th pime number:
Prime[1]=2
Prime[2]=3
Prime[3]=5
...

krassi_holmz · 2006-02-28 06:17:14

And for rob-ya, it's actually a great idea to call it "Krassi's number". But it's not os scientific as "Geogiev's constant".

John E. Franklin · 2006-03-01 07:33:03

Okay, now the program makes sense, thanks a lot.

God · 2006-03-02 11:17:52

Maybe the solution lies in the infinite pi-notation format of the reimann zeta function
After all, problems are a lot cooler to solve by hand than to approximate with computer

Last edited by God (2006-03-02 11:18:20)

krassi_holmz · 2006-03-03 00:15:49

What is that infinite pi-notation?
Can you explain it...
Simply?

John E. Franklin · 2015-04-04 06:43:55

can.t locate bug in k.h. program.
but pi squrd over six is just shy of ten over six.

bobbym · 2015-04-04 09:03:34

Hi;

There seems to be a slight inaccuracy in the determination of that constant. Correct to 100 places is

0.4522474200410654985065433648322479341732313432398924217364189303511650273639108744489575443549068582...

Math Is Fun Forum

#1 2006-02-18 22:44:56

Sum of the reciprocals of the squares of the prime numbers

#2 2006-02-18 22:46:53

Re: Sum of the reciprocals of the squares of the prime numbers

#3 2006-02-18 22:48:35

Re: Sum of the reciprocals of the squares of the prime numbers

#4 2006-02-18 22:51:33

Re: Sum of the reciprocals of the squares of the prime numbers

#5 2006-02-18 22:52:49

Re: Sum of the reciprocals of the squares of the prime numbers

#6 2006-02-20 09:09:29

Re: Sum of the reciprocals of the squares of the prime numbers

#7 2006-02-22 02:54:48

Re: Sum of the reciprocals of the squares of the prime numbers

#8 2006-02-22 05:55:10

Re: Sum of the reciprocals of the squares of the prime numbers

#9 2006-02-22 09:15:05

Re: Sum of the reciprocals of the squares of the prime numbers

#10 2006-02-28 06:15:37

Re: Sum of the reciprocals of the squares of the prime numbers

#11 2006-02-28 06:17:14

Re: Sum of the reciprocals of the squares of the prime numbers

#12 2006-03-01 07:33:03

Re: Sum of the reciprocals of the squares of the prime numbers

#13 2006-03-02 11:17:52

Re: Sum of the reciprocals of the squares of the prime numbers

#14 2006-03-03 00:15:49

Re: Sum of the reciprocals of the squares of the prime numbers

#15 2015-04-04 06:43:55

Re: Sum of the reciprocals of the squares of the prime numbers

#16 2015-04-04 09:03:34

Re: Sum of the reciprocals of the squares of the prime numbers

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